I've never been very comfortable with this notation for matrix groups; but I cannot see anything wrong with extending it to degenerate forms, apart from the fact that for some reason it's just not "standard". So I will just define O(p, q, r) == real (or whatever) n x n matrices preserving (x_1)^2 + ... + (x_p)^2 - (x_(p+1))^2 - (x_(p+q))^2 , where n = p+q+r . Now the usual representation of Euclidean 3-space isometries by homogeneous projective matrices becomes E(3) = PO(3, 0, 1) --- not as earlier botched --- and the Poincaré / Laguerre 3-space group would be PO(3, 1, 1) . Presumably this representation is well-known, though I have not seen it mentioned. Speaking of botches --- I managed to gain a extra space dimension earlier, my muddle arising from Mob(2) acting on the unit 2-sphere in 3-space. Should have read (groan) Lorentz == 2-space Möbius = Mob(2) = PO(3, 1) , dimension 6 ; Poincaré == 3-space Laguerre = Lag(3) = PO(3, 1, 1) , dimension 10 . WFL